Analytic time-dependent solutions of the one-dimensional Schrödinger equation

Dijk, W. van; Toyama, F. M.; Prins, Sjirk Jan; Spyksma, Kyle

doi:10.1119/1.4885376

Cited by 14 publications

(15 citation statements)

References 23 publications

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“…This result has been found by direct integration over the propagator. [2] The asymptotic form of free evolution as t → ∞ is determined by the Fourier transform of the initial wave function. [5] Thus…”

Section: Evolution Of Pure 1d Splinesmentioning

confidence: 99%

“…There are very few wave functions with compact support for which the exact evolution is known, either for a free particle or for a harmonic oscillator. One known case is the square wave function [2], where the evolution has been expressed in terms of error functions with complex argument (which can also be expressed in terms of real Fresnel integrals), using the integral over the propagator; but this wave function has infinite mean energy. The simple procedure discussed here yields the evolution not only of the square wave function but also a wide class of more physically realistic wave functions.…”

Section: Introductionmentioning

confidence: 99%

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The evolution of piecewise polynomial wave functions

Andrews

2017

Eur. Phys. J. Plus

161

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For a non-relativistic particle, we consider the evolution of wave functions that consist of polynomial segments, usually joined smoothly together. These spline wave functions are compact (that is, they are initially zero outside a finite region), but they immediately extend over all available space as they evolve. The simplest splines are the square and triangular wave functions in one dimension, but very complicated splines have been used in physics. In general the evolution of such spline wave functions can be expressed in terms of antiderivatives of the propagator; in the case of a free particle or an oscillator, all the evolutions are expressed exactly in terms of Fresnel integrals. Some extensions of these methods to two and three dimensions are discussed.

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Section: Evolution Of Pure 1d Splinesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The evolution of piecewise polynomial wave functions

Andrews

2017

Eur. Phys. J. Plus

161

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“…Given the recent attention to the two-dimensional nonlinear Schrödinger equation and the Pitaevskii equation, a future project is to expand the method of this paper to such systems, as well as those with time-dependent interactions or source terms. This in effect is a generalization of the work done earlier on one-dimensional systems [28].…”

Section: Discussionmentioning

confidence: 56%

“…An analytic solution for such a potential is [28] Ψ(x, y, t) = ψ nx (α x , β x ; x, t)ψ ny (α y , β y ; y, t), (4.11) where…”

Section: A Errorsmentioning

confidence: 99%

Numerical solutions of the time-dependent Schrödinger equation in two dimensions

2017

Self Cite

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The generalized Crank-Nicolson method is employed to obtain numerical solutions of the two-dimensional time-dependent Schrödinger equation. An adapted alternating-direction implicit method is used, along with a high-order finite difference scheme in space. Extra care has to be taken for the needed precision of the time development. The method permits a systematic study of the accuracy and efficiency in terms of powers of the spatial and temporal step sizes. To illustrate its utility the method is applied to several two-dimensional systems.

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“…We thus construct from two line integrals, each similar to the one appearing in the polar representation (4). As with the latter case, the integrals in (32) are action functions for a particle of mass m , but now moving in the potential Q + + V or Q − + V . Choosing suitable integration paths, each of the line integrals (for = ± ) may be written in either of the forms (7) or (8).…”

Section: Eliminating the Wavefunction From Quantum Dynamics: A Bi-tra...mentioning

confidence: 99%

Eliminating the Wavefunction from Quantum Dynamics: The Bi-Hamilton–Jacobi Theory, Trajectories and Time Reversal

Holland

2022

Found Phys

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We observe that Schrödinger’s equation may be written as two real coupled Hamilton–Jacobi (HJ)-like equations, each involving a quantum potential. Developing our established programme of representing the quantum state through exact free-standing deterministic trajectory models, it is shown how quantum evolution may be treated as the autonomous propagation of two coupled congruences. The wavefunction at a point is derived from two action functions, each generated by a single trajectory. The model shows that conservation as expressed through a continuity equation is not a necessary component of a trajectory theory of state. Probability is determined by the difference in the action functions, not by the congruence densities. The theory also illustrates how time-reversal symmetry may be implemented through the collective behaviour of elements that individually disobey the conventional transformation ($$\mathrm{T}$$ T ) of displacement (scalar) and velocity (reversal). We prove that an integral curve of the linear superposition of two vectors can be derived algebraically from the integral curves of one of the constituent vectors labelled by integral curves associated with the other constituent. A corollary establishes relations between displacement functions in diverse trajectory models, including where the functions obey different symmetry transformations. This is illustrated by showing that a ($$\mathrm{T}$$ T -obeying) de Broglie-Bohm trajectory is a sequence of points on the (non-$$\mathrm{T}$$ T ) HJ trajectories, and vice versa.

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Analytic time-dependent solutions of the one-dimensional Schrödinger equation

Cited by 14 publications

References 23 publications

The evolution of piecewise polynomial wave functions

The evolution of piecewise polynomial wave functions

Numerical solutions of the time-dependent Schrödinger equation in two dimensions

Eliminating the Wavefunction from Quantum Dynamics: The Bi-Hamilton–Jacobi Theory, Trajectories and Time Reversal

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